9

Correlations and Something About a Freezer and an Oven¹

Let’s not forget that the quality of a model must be compared with another model. The best model is the one that leads to better decisions.

—JPP

WHEN WE FORECAST RISK AT THE PORTFOLIO LEVEL, WE must estimate correlations, either directly or indirectly. The concept of diversification is at the core of every asset allocation decision. It’s the key premise and goal of portfolio construction. The lower the correlations, the better the diversification, and if we hold everything else constant, the higher the risk-adjusted return.

Yet one of the most vexing problems in investment management is that diversification seems to disappear when investors need it the most. Of course, the statement that “all correlations go to 1 in a crisis” is both an oversimplification and an exaggeration. But it has been well documented that correlations tend to increase in down markets, especially during crashes (i.e., “left-tail events”). Studies have shown that this effect is pervasive for a large variety of financial assets, including individual stocks, country equity markets, global equity industries, hedge funds, currencies, and international bond markets.² Most of these studies were published before the 2008 global financial crisis. Yet the failure of diversification during the crisis, when left-tail correlations jumped significantly, seemed to surprise investors.

In a 2018 paper published in the Financial Analysts Journal, “When Diversification Fails,” Rob Panariello and I encouraged investors to act on such findings. Full-sample (i.e., average) correlations are misleading. Prudent investors should not use them in risk models, at least not without adding other tools, such as downside risk measures and scenario analyses. To enhance risk management beyond naïve diversification, investors should reoptimize portfolios with a focus on downside risk, consider dynamic strategies, and, depending on aversion to losses, evaluate the value of downside protection as an alternative to asset class diversification.

The Myth of Diversification

Back in 2008, Charlie Henneman, Head of Educational Events and Programs at the CFA Institute, asked me to present at the Institute’s Annual Risk Management Conference.

Charlie asked for my talk’s title.

At the time, I had started to work on the instability of correlations. I suggested a wonky title: “Asymmetries in Cross-Asset Conditional Correlations.”

Charlie didn’t want a technical title. Who wants to attend a talk titled “Asymmetries in Cross-Asset Conditional Correlations”? Might as well call it “Another Obscure Academic Talk with Little Relevance to Investors.”

I told him that essentially, I would show that diversification doesn’t work when we need it the most. He then suggested a fantastic title: “The Myth of Diversification.”

The talk was a success, as was a paper of the same title, written with David Chua and Mark Kritzman (2009), which won the 12th Annual Bernstein Fabozzi/Jacobs Levy Outstanding Article Award. I’m convinced the title helped. The timing of the publication was fortuitous too: we used a precrisis data sample ending in February 2008 and documented significant “undesirable correlation asymmetries” for a broad range of asset classes. The events of the fall of 2008 reinforced our results.

We added another angle (as usual, Mark’s idea): we showed that not only did correlations increase on the downside, but they also significantly decreased on the upside. This asymmetry is the opposite of what investors want. Indeed, who wants diversification on the upside? Upside unification (or antidiversification) would be preferable. During good times, we should seek to reduce the return drag from diversifiers.

In our more recent (2018) paper, Rob Panariello and I insist that despite the wide body of published research, many investors still do not fully appreciate the impact of correlation asymmetries on portfolio efficiency—in particular, on exposure to loss. During left-tail events, diversified portfolios may have greater exposure to loss than more concentrated portfolios. Marty Leibowitz and Anthony Bova (2009) show that during the 2008 global financial crisis, a portfolio diversified across US stocks, US bonds, international stocks, emerging markets stocks, and REITs saw its equity beta rise from 0.65 to 0.95, and the portfolio unexpectedly underperformed a simple 60% US stocks/40% US bonds portfolio by 9 percentage points. Similar effects were just observed during the crisis of 2020.

Rob and I expanded the analysis of the original “Myth of Diversification” in several ways. We included post-2008 data, we covered a broader set of markets and factors, and we took an in-depth look at what drives correlations in numerous markets. As for methodology, we introduced a data-augmentation technique to improve the robustness of tail correlation estimates, and we analyzed the impact of return data frequency on private asset correlations.

How correlations change during extreme markets can be estimated in several ways. For example, we can identify months during which both assets moved (up or down) by at least a given percentage, which is called “double conditioning.”³ Rob and I used a similar approach, but we conditioned on a single asset. We wanted to measure differences in tail correlations based on which of the two markets sold off (or rallied). For some correlations, such as the stock-bond correlation, this difference can be substantial, and it adds information on the correlation structure.

We wanted to evaluate the effectiveness of bond diversification during US stock market sell-offs (the flight-to-safety effect). First, we isolated months in our data sample during which US stocks were down significantly: say, their worst 5% of all monthly returns in our sample. Next, we calculated a correlation between stocks and bonds in this subsample. Then we moved the percentile cutoff by a fixed increment to see how the correlation changes as a function of stock returns.

We also calculated the stock-bond correlation when bonds were in their bottom 5%. We found that bonds diversify stocks when stocks sell off, but stocks do not diversify bonds when bonds sell off. Double conditioning would fail to reveal this lack of symmetry in the diversification between the two assets.

Irrespective of how we partitioned the data, we expected subsample correlations to differ from full-sample estimates. To measure this “conditioning bias,” we first simulated how correlations change when we move toward the left and right tails (from a standard “bivariate normal distribution”) based on random data. For each asset pair, we simulated two normal distributions with the same full-sample correlations, means, and volatilities as those we observed empirically.

Then we compared the actual subsample correlations with their simulated normal counterparts. Differences indicated departures from normality. Also, under normality, downside and upside correlation profiles should be identical. Therefore, when we compared left-tail and right-tail correlations, the conditioning bias did not matter much because it “washed out.” Any asymmetry we found indicated a departure from normality.

Another possible bias arises because extreme correlations rely on few data points, an issue similar to the estimation of higher moments I mentioned earlier. The further one goes into the tails, the smaller the sample. At the top or bottom 1 or 5% of the distribution, a single outlier may significantly bias correlations up or down. To increase robustness in our estimates, therefore, we augmented subsamples with data from the rest of the distribution.

We used an exponentially weighted approach. The closest the additional data points were to the cutoff of interest, the more weight they received in our estimation. These weights decreased exponentially as we moved further from the threshold. This approach, although simple and intuitive, had not been used in prior studies; hence, perhaps we made a modest methodological contribution to the measurement of conditional correlations. (To be more precise, we calibrated the model in such a way that observations further into the tails received exponentially larger weights, and we fixed the half-life at the percentile under consideration.⁴)

For comparison, we also reported unadjusted conditional correlations. We found that the data-augmentation methodology generated estimates similar to those calculated conventionally, in terms of magnitude and directionality. But our estimates tended to be less noisy, and they were less sensitive to outliers.⁵

The Failure of Diversification Across Risk Assets

First, we looked at international equity diversion to illustrate our approach. Based on monthly data from January 1970 to June 2017, we calculated conditional correlations between US stocks (MSCI U.S. Total Return Index) and non-US stocks (MSCI EAFE Total Return Index). We conditioned correlations by percentile, based on the returns of US stocks. Our results showed how correlations changed from the worst sell-offs in US stocks (1st percentile) to their strongest rallies (99th percentile). For comparison, we showed the correlation profile that we would expect if both markets were normally distributed. In the normal case, we would expect perfect symmetry between upside and downside correlations. Conditional correlations would gradually decrease as we moved toward the tails.

Real-world correlations differed substantially from their normally distributed counterparts. When US stocks were rallying (in their 99th percentile), their correlation with non-US stocks dropped all the way to –17%. During the worst sell-offs, represented by the bottom 1% of all returns in US stocks, however, their correlation with non-US stocks rose to +87%. This asymmetry revealed that international diversification works only on the upside.⁶

We found similar results across risk assets. In Table 9.1, I show a comparison of left-tail and right-tail correlations for key asset classes, based on available data histories as of June 2017.⁷ The focus is on US stocks versus other risk assets because the equity risk factor dominates the volatility factor (and exposure to loss) in most portfolios.⁸ We used bond returns net of duration-matched US Treasuries (i.e., “excess returns”) to isolate credit risk factors.

We also show results for style and size diversification within stocks. Most investors select equity funds—and thereby seek to diversify their portfolios—based on style/size characteristics. Across the board, left-tail correlations are much higher than right-tail correlations.

TABLE 9.1 Extreme Correlations

Studies on “tail dependence” (how crashes tend to happen at the same time across markets) corroborate these findings. Garcia-Feijoo, Jensen, and Johnson (2012) show that when US equity returns are in their bottom 5%, non-US equities, commodities, and REITs also experience significantly negative returns—beyond what would be expected from full-sample correlations. Hartmann, Straetmans, and de Vries (2010) show that currencies co-crash more often than would be predicted by a bivariate normal distribution. Hartmann, Straetmans, and de Vries (2004) estimate that stock markets in G5 countries are two times more likely to co-crash than bond markets.

Van Oordt and Zhou (2012) extend pairwise analysis to joint tail dependence across multiple markets and reach similar conclusions. They suggested a related approach to measure the systemic importance of financial institutions. These studies ignored asymmetries, however, between the left and right tails. They either focused on the left tail or used symmetrical measures of tail dependence, such as the joint t-distributions.

Regarding credit asset classes, the Merton (1974) model explains why credit and equity returns become more correlated in the left tail. Merton defined a corporate bond as a combination of:

  A risk-free bond—in normal times, the bondholders’ upside risk is limited to the regular coupon payments and return of principal.

  A short put position on the company’s assets. If the company’s asset value depreciates below its debt, bondholders become long the company’s assets and receive what’s left through bankruptcy proceedings. (Meanwhile, as the stock price goes to zero, stockholders are wiped out.)

Hence, as a company approaches default, the market starts to expect that bondholders will be left holding the bag (of the company’s remaining assets). Merton explained that “as the probability of eventual default becomes large, . . . the risk characteristics of the debt approach that of (unlevered) equity.” In this context, it was not a surprise that during the 2008 crisis, credit and equity returns became highly correlated.⁹

Diversification fails across styles, sizes, geographies, and alternative assets. Essentially, all the return-seeking building blocks that asset allocators typically use for portfolio construction are affected. The asymmetry for the stock-MBS (mortgage-backed securities) correlation is notable. In “The Myth of Diversification” paper (2009), we had used precrisis data, and at the time of the study, MBS was one of the few asset classes that seemed to decouple from stocks in down markets. During the fourth quarter of 2008, however, which is included in this data sample, MBS clearly joined the ranks of risk-on assets.

Alternative assets are not immune to the failure of diversification. Beyond traditional asset classes, investors have increasingly looked to alternatives for new or specialized sources of diversification. In Table 9.1, Rob and I used a broad hedge fund index, but one could argue that hedge fund styles are so different from each other that they should be treated as separate asset classes. We decided to compare left-tail and right-tail correlations (versus US stocks) for seven hedge fund styles: equity market neutral, merger arbitrage, event driven, macro, equity hedge, relative value convertible, and relative value. Unfortunately, we found that all these types of hedge funds, including the market-neutral funds, exhibited significantly higher left-tail than right-tail correlations. While the average right-tail correlation was –7%, the average left-tail correlation jumped to +63%.¹⁰

A simple explanation could be that most hedge fund strategies are short volatility. Some are also short liquidity risk, which is akin to selling an option, as explained by my former PIMCO colleague Vineer Bhansali (2010) in his book Bond Portfolio Investing and Risk Management. Agarwal and Naik (2004) explain jumps in hedge fund left-tail equity betas through the Merton (1974) lens. They observed that “a wide range of hedge fund strategies exhibit returns similar to those from writing a put option on the equity index.” In a related study, Billio, Getmansky, and Pelizzon (2012) use a regime-switching model to measure hedge fund correlations and market betas over time. They show that the average jump in correlations for hedge fund strategies in financial crises was +33%.

What about private assets? Over the past few years, institutional investors have significantly increased their allocations to private assets. Although many investors have become skeptical of the diversification benefits of hedge funds, the belief in the benefits of direct real estate and private equity diversification has been persistent. The advisory firm Willis Towers Watson reports that as of the end of 2016, pension funds, wealth managers, and sovereign wealth funds held more than $2 trillion in direct real estate and private equity investments.¹¹ Money has flowed into these asset classes partly because of their perceived diversification benefits. Consultants have used mean-variance optimization in asset allocation or asset liability studies to make a strong case for increased allocations. In the end, alternative assets are often sold as free lunches because they seem to offer high returns with low volatility and great diversification properties.

But there is more to these statistics than meets the eye. Private assets’ reported returns suffer from the smoothing bias. In fact, in a paper my former colleagues Niels Pedersen, Fei He, and I published in the Financial Analysts Journal in 2014, titled “Asset Allocation: Risk Models for Alternative Investments” (which won a Graham and Dodd Award for Excellence in Research), we show that private assets’ diversification advantage is almost entirely illusory. We argue that reported quarterly returns for private assets represent a moving average of the true (unobserved) marked-to-market returns. On a marked-to-market basis, these asset classes are exposed to many of the same factors that drive stock and bond returns. For example, and as we’ve shown, after their smoothing bias is removed, private assets have exposure to credit risk, which does not truly diversify equity risk in times of market stress.

Even David Swensen, chief investment officer of Yale University’s Endowment (which has a large proportion of its assets invested in private markets), acknowledges the issue in his 2009 book, Pioneering Portfolio Management:

The low risk evident in data describing past returns from private investing constitutes a statistical artifact. . . . If two otherwise identical companies differ only in the form of organization—one private, the other public—the infrequently valued private company appears much more stable than the frequently valued publicly traded company.¹²

From Pioneering Portfolio Management by David F. Swensen. Copyright © 2000 by David F. Swensen. Reprinted with the permission of The Free Press, a division of Simon & Schuster, Inc. All rights reserved.

Not only is the true equity risk exposure of private assets higher than implied by their reported returns on average, but their left-tail exposures are much higher. To illustrate, in “When Diversification Fails,” Rob and I showed results from a test that any investor can easily replicate. For direct real estate and private equity, we compared quarterly to rolling annual (four quarters) left-tail correlations with equity. (Rolling annual correlations are less sensitive to the smoothing bias than those calculated on quarterly returns.)

This remarkably simple adjustment revealed that private real estate and private equity don’t diversify equity risk as much as most investors assume. Although the quarterly correlation between real estate and US stocks was +29%, it jumped to +67% on a rolling annual basis. For private equity, the quarterly correlation was +13%, compared with +85% on a rolling annual basis.¹³

In addition to the smoothing bias, we surmised that liquidity risk contributed to the failure of private asset diversification. Because they don’t trade as frequently, private assets are exposed to significant liquidity risk, perhaps even more than hedge funds. Systemic liquidity risk tends to manifest itself during stock market crashes.¹⁴ A systemic liquidity crisis can be compared with a burning building, in which everyone is rushing for the door, with one exception: In financial markets, to get out (sell), investors must find someone to take their place in the building (a buyer). When no buyers are present, prices crash instantaneously, and correlations across risk assets jump.

What about risk factors? The failure of diversification across public and private return-seeking asset classes has led, in part, to the popularity of risk factors. Many authors have argued that risk factor diversification is more effective than asset class diversification.¹⁵ Our results indeed revealed that several risk factors (equity value, cross-asset value, equity momentum, currency value, and currency momentum) appear to be more immune to the failure of diversification than are asset classes.

Others have pointed out, however, that risk factors aren’t inherently superior building blocks.¹⁶ They deliver better diversification than traditional asset classes simply because they allow short positions and often encompass a broader universe of assets. For example, the size and value factors in equities are often defined as long-short, security-level portfolios. But if factor definitions are restricted to linear combinations of asset classes, and short positions are allowed across asset classes as well as risk factors, then risk factors do not deliver any efficiency gains over asset classes. In a sense, the argument in favor of risk factor diversification is more about the removal of the long-only constraint and the expansion of the investment universe than anything else.

In addition, momentum strategies that sell risk assets in down markets provide left-tail diversification, almost by definition. Portfolio insurance strategies, for example, can explicitly replicate a put option (minus the gap-risk protection). Hence, as expected, we found that risk factors such as currency and cross-asset momentum had much lower left-tail than right-tail correlations with US stocks.

However, our results also showed that risk-on factors, such as size (i.e., small minus big stocks) and currency carry, may fail to diversify stocks when needed. Small cap stocks tend to have higher equity betas than large cap stocks, and this difference in market beta exposure is often expressed during stock market drawdowns. Similarly, the currency carry trade has an indirect equity beta exposure that remains dormant until risk assets sell off. The strategy is to buy high-interest-rate currencies (the Australian dollar, emerging markets currencies, etc.) and to fund these positions by shorting low-interest- rate currencies (for example, the Japanese yen).

In normal markets, the investor earns a risk premium because forward rates typically do not appreciate or depreciate enough to offset profits (the carry) from the interest rate differential embedded in currency forward contracts. But when risk assets sell off, the carry trade unwinds as investors sell the higher-risk currencies and buy the safe havens. In a sense, many carry strategies behave like the credit risk premium. These strategies are short an option, and investors sometimes refer to the adage “Picking up pennies in front of a steamroller” to describe them.

The example of the currency carry trade illustrates the impact of regime shifts on correlations, which may explain the widespread risk-on/risk-off characteristic of return-seeking asset classes and risk factors. Financial markets tend to fluctuate between a low-volatility state and a panic-driven, high-volatility state (see, for example, Kritzman, Page, and Turkington, 2012). In fact, Ang and Bekaert (2002) directly link the concept of regime shifts to rising left-tail correlations. But what causes regime shifts? A partial answer is that macroeconomic fundamentals themselves exhibit regime shifts, as documented for inflation and growth data.¹⁷

I think investor sentiment also plays a large role. In normal markets, differences in fundamentals drive diversification across risk assets. During panics, however, investors often “sell risk” irrespective of differences in fundamentals. Huang, Rossi, and Wang (2015), for example, show that sentiment is a common factor that drives both equity and credit-spread returns—beyond the effects of default risk, liquidity, and macro variables—and suggest that sentiment often spills over from equities to the credit markets.

In financial markets, fear is more contagious than optimism. Related studies in the field of psychology suggest that to react more strongly to bad news than good news is human nature. In their paper “Bad Is Stronger Than Good,” Baumeister, Bratslavsky, Finkenauer, and Vohs (2001) explain:

The greater power of bad events over good ones is found in everyday events, major life events, close relationship outcomes, social network patterns, interpersonal interactions, and learning processes. . . . Bad information is processed more thoroughly than good. . . . From our perspective, it is evolutionarily adaptive for bad to be stronger than good.

Is the Stock-Bond Correlation the Only True Source of Diversification?

When market sentiment suddenly turns negative and fear grips markets, government bonds almost always rally because of the flight-to-safety effect.¹⁸ In a sense, duration risk may be the only true source of diversification in multi-asset portfolios. Therefore, the stock-bond correlation is one of the most important inputs to asset allocation.

In our study, we compared the empirical stock-bond conditional correlation profile with its normally distributed benchmark. Unlike results for other correlations, this profile was highly desirable. Treasuries decouple from stocks in bad times and become positively correlated with stocks in good times. The stock-bond correlation is difficult to estimate, however, and can change drastically with macroeconomic conditions (Johnson, Naik, Page, Pedersen, and Sapra, 2014).

In a 2014 Journal of Investment Strategies paper titled “The Stock-Bond Correlation,” which I coauthored with my former PIMCO colleagues Nic Johnson, Vasant Naik, Niels Pedersen, and Steve Sapra, we demonstrate that when inflation and interest rates drive market volatility, the stock-bond correlation often turns positive. For example, we show that the 12-month stock-bond correlation was mostly positive during the 1970s and 1980s. Since 2008, central bank stimulus and declining rates have artificially pushed valuations higher in both the stock and bond markets. This type of “sugar high” can unwind quickly if policy normalizes unexpectedly. The “taper tantrum” of 2013, when Ben Bernanke first mentioned the idea of tapering the Fed’s stimulus, provides a good example. It affected stocks and bonds negatively at the same time.

More recently, in 2018, when the Federal Reserve hiked its target rate from 2% to 2.5% and signaled further hikes in 2019, both stocks and bonds suffered negative returns over the entire calendar year, which is rare. In fact, all the 17 major asset classes that Morgan Stanley tracks were down in 2018, an unprecedented outcome.¹⁹ So much for asset class diversification. Investors should remember that starting valuations can compound the effect. The higher the valuations in both stocks and bonds, the more fragile their correlation.

To illustrate how bond sell-offs can lead to a positive stock-bond correlation, we estimated the stock-bond correlation as a function of percentiles in bond returns instead of stock returns. This change in methodology revealed an interesting pattern. The correlation profile was not as desirable as when we conditioned on stock returns. Although the correlations were generally low, when bonds sell off, stocks tend to sell off at the same time. Ultimately, investors should remember that stocks and bonds both represent discounted cash flows. Unexpected changes in the discount rate or inflation expectations can push the stock-bond correlation into positive territory—especially when other conditions remain constant.

Despite consistent results across markets, there are caveats to our study. We showed that during crises, diversification across risk assets almost always fails, and even the stock-bond correlation may fail in certain market environments. But conditional correlations represent only one way to measure diversification. Conditional betas, for example, take into account changes in relative volatilities as well as correlations. In theory, it is possible for the correlation between two assets to increase while the volatility of the diversifier decreases relative to the main engine of portfolio growth. In this case, a correlation spike may be offset by decreasing relative volatilities, which could lead to a lowered stress beta and, perhaps, lower exposure to loss than expected. However, prior studies have shown that such outcomes are highly improbable.²⁰ Ultimately, we chose to study correlations. They measure diversification directly and have been used widely in earlier studies.

Another caveat is that we did not forecast left-tail events. Although we know that correlations are likely to increase if markets sell off, we do not necessarily know when this shift will take place. Equity sell-offs are unexpected, almost by definition. Investors can prepare for the failure of diversification, however, without the need to time markets. Consider as an analogy that although it is almost impossible for aircraft pilots to predict when they will encounter air turbulence, passengers can take comfort in the fact that airplanes are built to withstand it.

Recommendations for Asset Allocation and Tail-Risk-Aware Analytics

In light of our results, and as reinforced by the events of 2020, I recommend that investors avoid the use of full-sample correlations in portfolio construction—or, at least, that they stress-test their correlation assumptions. Scenario analysis, either historical or forward-looking, should take a bigger role in asset allocation than it does. A wide range of portfolio optimization methodologies directly address nonnormal left-tail risk and, ipso facto, the failure of diversification. The most flexible is full-scale optimization, which optimizes directly on the empirical distribution of returns, for any type of investor preferences and goals.²¹ (We discuss full-scale optimization in Chapter 14.)

These analytics are commonly used, but almost always on a “posttrade” basis—that is, after portfolio construction has taken place. Investors should use such tail-aware tools as part of “pretrade” decisions. To do so will reveal that equity regions, styles, sizes, and sectors—as well as credit, alternative assets, and risk factors—do not diversify broad equity risk as much as average correlations suggest. To be clear, I’m not arguing against diversification across traditional asset classes, but investors should be aware that traditional measures of diversification may belie exposure to loss in times of stress. Investors should calibrate their risk tolerance accordingly against return opportunities.

In addition, beware the stock-bond correlation. Shocks to interest rates or inflation can turn it positive. In such situations, strategies that use leverage to increase the contribution to the risk of bonds—risk parity, for example—may experience unexpected drawdowns.

Finally, investors should look beyond diversification to manage portfolio risk. Tail-risk hedging (with equity put options or proxies), risk factors that embed short positions or defensive momentum strategies, and dynamic risk-based strategies all provide better left-tail protection than traditional diversification.

The strategy of managed volatility can be a particularly effective and low-cost approach to overcome the failure of diversification. As we discussed in Chapter 7, based on the empirical observation that risk is more predictable than return, this strategy adjusts the asset mix over time to stabilize a portfolio’s volatility. It is portable and can easily be applied as an overlay to smooth the ride for almost any portfolio. Because managed volatility will scale down risk assets when volatility is high, it may offset left-tail correlation spikes and thereby reduce exposure to large losses without a significant reduction in returns on the upside.

As part of preparing to present this study at a conference for Canadian institutional investors, I formulated seven key recommendations for tail-risk-aware analytics and suggested six active management strategies. The tail-risk-aware analytics are necessary to evaluate the effectiveness of the active management strategies and to scale exposures to them. The items below sum up our earlier discussion in this chapter:

These analytics and active management strategies are now widely available to help investors manage the failure of diversification.

To summarize the issue Rob and I highlighted in our paper, I often use the story of the statistician who had his head in the oven and his feet in the freezer. He suddenly exclaimed, “On average, I feel great!” (Straight from my repertoire of cheesy conference jokes. This one gets polite chuckles, at best.)

Similarly, as a measure of diversification, the full-sample correlation is an average of extremes. During market crises, diversification across risk assets almost completely disappears. Moreover, diversification seems to work remarkably well when investors do not need it: during market rallies. This undesirable asymmetry is pervasive across markets.

Notes

1.   This section is largely taken from “When Diversification Fails,” a recent paper Rob Panariello and I published in the Financial Analysts Journal. See Page and Panariello (2018).

2.   See, for example, Ang, Chen, and Xing (2002), Ang and Chen (2002), and Hong, Tu, and Zhou (2007) on individual stocks; Longin and Solnik (2001) on country equity markets; Ferreira and Gama (2010) on global industries; Van Royen (2002b) and Agarwal and Naik (2004) on hedge funds; Hartmann, Straetmans, and de Vries (2010) on currencies; and Cappiello, Engle, and Sheppard (2006) on international equity and bond markets.

3.   See, for example, Longin and Solnik (2001) and Chua, Kritzman, and Page (2009).

4.   “Half-life,” in this context, means the point at which the sum of the weights reaches 50% (out of 100%).

5.   An important point regarding the conditioning bias is that we applied the same exponential adjustment to the corresponding simulated normal data. Hence, in all cases, comparisons between empirical and normal correlations were apples to apples.

6.   Longin and Solnik (2001), focusing on the correlations between the United States, France, Germany, the United Kingdom, and Japan, reported similar results for stocks at the country level.

7.   In the table, EM is “emerging market.” Monthly data, with start dates based on availability. See Appendix B from “When Diversification Fails” (Page and Panariello, 2018), available online at http://www.cfapubs.org/doi/suppl/10.2469/faj.v74.n3.3, for start dates and data sources. Left-tail and right-tail correlations are at the 1st and 99th percentiles but were adjusted by the data-augmentation methodology. Full correlation profiles (adjusted, unadjusted, and normal) are shown in Appendix B.

8.   See, for example, Page (2013).

9.   See, for example, Naik, Devarajan, Nowobilski, Page, and Pedersen (2016).

10.   See Page and Panariello (2018) for more details on the data and methodology.

11.   As reported in the Financial Times, Chris Flood, “Global Shift into Alternatives Gathers Pace,” July 16, 2017, http://www.ft.com.

12.   David Foulke (2016) highlighted this quote on Alpha Architect.

13.   Full correlation profiles available online at https://www.cfapubs.org/doi/suppl/10.2469/faj.v74.n3.3.

14.   See Page, Simonian, and He (2011).

15.   See, for example, Bender, Briand, Nielsen, and Stefek (2010); Page and Taborsky (2011): and Ilmanen and Kizer (2012).

16.   See Idzorek and Kowara (2013) and Cocoma, Czasonis, Kritzman, and Turkington (2017).

17.   See, for example, Kim (1993) and Kumar and Okimoto (2007) on inflation and Hamilton (1989), Goodwin (1993), Luginbuhl and de Vos (1999), and Lam (2004) on GDP/GNP growth.

18.   See, for example, Gulko (2002).

19.   For the period during which data are available. Source: Morgan Stanley, monthly data pack, “Global in the Flow,” 2018.

20.   See, for example, Leibowitz and Bova (2009) on betas, Hartmann, Straetmans, and de Vries (2004, 2010) on co-crash probabilities, and Garcia-Feijoo, Jensen, and Johnson (2012) on tail dependence.

21.   See, for example, Cremers, Kritzman, and Page (2005), Sharpe (2007), and Adler and Kritzman (2007).